Edit face triangulation without actually adding edge
Triangulation can't be controlled directly, it's not even done by Blender in object mode. Instead, it is left to the graphics driver to decide how to triangulate non-planar quads (which differs among drivers for different cards). Behavior may even change with VBO enabled or disabled.



You can use the triangulation modifier to get a bit of control with the beauty option, to divide across the shortest diagonal: